Harman Prime-Detecting Sieves (LMS-33)

Course Book
ISBN: 978-1-4008-4593-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 384 Seiten

Reihe: London Mathematical Society Monographs

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec. Prime-Detecting Sieves breaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information.

No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube. This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.

Autoren/Hrsg.

Harman, Glyn

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

Preface xi

Notation xiii

Chapter 1. Introduction 1

Chapter 2. The Vaughan Identity 25

Chapter 3. The Alternative Sieve 47

Chapter 4. The Rosser-Iwaniec Sieve 65

Chapter 5. Developing the Alternative Sieve 83

Chapter 6. An Upper-Bound Sieve 103

Chapter 7. Primes in Short Intervals 119

Chapter 8. The Brun-Titchmarsh Theorem on Average 157

Chapter 9. Primes in Almost All Intervals 189

Chapter 10. Combination with the Vector Sieve 201

Chapter 11. Generalizing to Algebraic Number Fields 231

Chapter 12. Variations on Gaussian Primes 265

Chapter 13. Primes of the Form x3 + 2y3 303

Chapter 14. Epilogue 335

Appendix 337

Bibliography 349

Index 361

Über Autor(innen)

Glyn Harman is professor of pure mathematics at the University of London, Royal Holloway. He is the author of Metric Number Theory, the coeditor of Sieve Methods, Exponential Sums, and their Applications in Number Theory, and the corecipient of the Hardy-Ramanujan award for his work on primes in short intervals.